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Register a new userNMR implementation of Factoring Large Numbers with Gauss{}Sums: Suppression of Ghost Factors
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Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al., which uses specific properties of Gauss{}sums. Experimental limitations usually require truncation of these series, but if the truncation parameter is too small, it is no longer possible to distinguish between factors and so-called ghost factors. Here, we discuss two techniques for distinguishing between true factors and ghost factors while keeping the number of terms in the sum constant or only slowly increasing. We experimentally test these modified algorithms in a nuclear spin system, using NMR.
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We construct an analog computer based on light interference to encode the hyperbolic function f({zeta}) = 1/{zeta} into a sequence of skewed curlicue functions. The resulting interferogram when scaled appropriately allows us to find the prime number decompositions of integers. We implement this idea exploiting polychromatic optical interference in a multipath interferometer and factor seven-digit numbers. We give an estimate for the largest number that can be factored by this scheme.
We study the theory of linearly chirped biphoton wave-packets produced in two basic quasi-phase-matching configurations: chirped photonic-like crystals and aperiodically poled crystals. The novelty is that these structures are considered as definite assembles of nonlinear layers that leads to detailed description of spontaneous parametric down-conversion (SPDC) processes through the discrete Gauss sums. We demonstrate that biphoton spectra for chirped photonic crystals involving a small number of layers consist from definite well-resolved spectral lines. We also discuss the forming of broadband spectra of signal (idler) waves in SPDC for both configurations as number of layers increases as well as in dependence of chirping parameters .
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